Quantum Origins of Magnetism

For most people, magnetism is something they first experience with everyday magnets — the ones that pick up paperclips or cling to a fridge. If you break a magnet in two, each piece becomes a smaller magnet. This raises a natural question: what is the fundamental building block of a magnet? Clearly, it’s not the piece of metal we see and touch, but something much smaller. This article presents to you how magnetism arises from the tiny electrons inside materials.

Solids are composed of atoms which contain sub-atomic particles called electrons. Turns out these electrons each are like bar magnets, their orientation represented by a vector called  “magnetic moment”. Mysteriously, this vector can only point in two directions, which is why physicists describe it using “spin”---taking the value ﹢½ when the vector points “up” and -½ when it points down.

Spins interact with each other in numerous ways, causing either parallel or antiparallel alignment. Whichever alignment minimizes the total energy of electrons is favored.

Heisenberg Exchange

Exchange essentially means “caused by overlap of wavefunctions".

One key insight from quantum mechanics is that, in the absence of external influences, electrons behave like waves. Their behavior is described by a wavefunction, which encodes the probability distribution of their characteristics. The wavefunction is found by solving the Schrödinger equation, and in many cases it can be written as a product of a spatial function and a spin function. These components describe, respectively, how an electron’s properties depend on its position and the direction of its magnetic moment.

When two electrons are considered together, their properties are described by a shared wavefunction. A fundamental requirement for electrons is that their total wavefunction changes sign when the two electrons are exchanged — that is, when all their quantum numbers are swapped. Physicists call this property “antisymmetry under exchange”.

For a system of two electrons, the collective spin function can be either symmetric or antisymmetric, leading to two classes of states: the singlet and the triplet. These two spin configurations correspond to different energies, typically denoted ES and ET. The total energy then becomes E=-ES-ET2(SiSj) , where Si and Sj are vectors in the direction of the magnetic moment. When ES>ET, parallel alignment is favored and when  ES<ET, antiparallel alignment is favored. 

Superexchange

Superexchange is the indirect exchange interaction between non-neighboring magnetic ions mediated by a non-magnetic ion placed between them. 

Within each atom, electrons sit in pairs in "orbitals". Their close proximity means that if they had the same spins, they’d be almost indistinguishable from each other, so get this: their wavefunction after exchange of electrons equals itself, which means that it’s 0! In other words, these electrons have zero probability of existing if they have the same spin in the same orbital.

Superexchange relies on the fact that the collective wavefunction of the ions’ electrons span the entire system (two magnetic ions+mediator), meaning that electrons can “hop” between ions. The mediator is chosen as non-magnetic so that it has no unpaired electron/unfilled orbital and thus no net magnetic moment. It follows that antiparallel spins in the magnetic ions make it easier for electrons to hop into the empty orbital of the mediator, and between ions. States in which electrons are more delocalized tend to have lower energy, thus, superexchange favors antiparallel alignment of magnetic moments. 

Intrinsic magnetism

Exchange interactions are the main reason why spins in magnetic materials have long-range order in the absence of an applied magnetic field. When parallel alignment is largely favored, the spins align and “combine to create a permanent, macroscopic magnet”. This phenomenon is called "ferromagnetism".

When antiparallel alignment is favored, neighboring spins point in opposite directions, effectively canceling each other out. As a result, no large-scale magnet forms and the phenomenon is called "antiferromagnetism".

Under an applied magnetic field, new interactions come into play…

Para- and Diamagnetism

Accounting for the applied field in the total energy of an electron reveals a paramagnetic and diamagnetic term. The former makes spins align with the field and the latter makes it antialign. However, we know that like a needle compass, spins tend to align with external magnetic fields; so why is there a diamagnetic contribution?

Classically, any moving electron’s motion is affected by an applied field such that its motion would create a magnetic field opposite to the applied one, according to a principle called Lenz’s law.  It is the electron’s “orbital motion” around an atom that produces this field, which in turn causes nearby spins to antialign with the applied field. The quantum mechanical analogue is a bit more abstract, but leads to the same result.

Though not as strong as exchange interactions, diamagnetism is more temperature-independent as the produced field  is present regardless of how an electron moves or how its spins are directed. Thermal excitations makes it difficult for electron spins to align/antialign in order to produce a “molecular field”, decreasing the effectiveness of exchange.

Temperature dependence

To produce a molecular field through paramagnetism, the spins must align under the applied field first. As temperature increases, spin directions become more “random”, making the applied field weaker.

For a ferromagnetic material, the produced field weakens until a critical temperature, called the “Curie temperature”, is reached. At this point, all long-range order suddenly vanishes as a phase transition occurs and the material becomes paramagnetic.

From exchange to para/diamagnetism to thermal fluctuations, magnetism is one of the many examples of how macroscopic properties arise from microscopic behaviour, described by the rules of quantum mechanics.

References:

Blundell, Stephen. Magnetism in Condensed Matter. Oxford University Press, 2001.